All Questions
Tagged with algebraic-k-theory at.algebraic-topology
95 questions
12
votes
1
answer
458
views
Algebraic K-theory of a ring
I started to learn some algebraic $K$-theory and its relation to geometric topology problems.
My question is: What is the list of rings such that all their algebraic $K$-theory groups are known?
I ...
- 433
12
votes
0
answers
551
views
Goodwillie's notes from MSRI Lecture Series
Does anyone know where I can find an electronic version of Goodwillie's (unpublished) notes from the MSRI Lecture Series in Spring, 1990? They're mentioned/cited as such in work of Dundas-McCarthy, ...
19
votes
2
answers
1k
views
Is there an ∞-categorical interpretation of the Quillen S⁻¹S construction?
The Quillen S⁻¹S construction (not to be confused with the Quillen Q-construction or the Quillen plus-construction),
as defined by Grayson in Higher algebraic K-theory: II (page 219),
takes as an ...
- 37.8k
6
votes
2
answers
1k
views
K theory long exact sequence
(1) Suppose that $Z\subset X$ is a closed embedding, $U = X\setminus Z$ is the complement. If relevant, suppose that both $X, Z$ are smooth and even (if relevant) that the normal bundle of $Z\subset X$...
- 7,952
5
votes
1
answer
300
views
Map between homotopy groups of O, related to J-homomorphism and K-theory of Z
Let $s \geq 0$ be fixed. The $J$-homomorphism includes $\pi_{8s+1}(SO) = \mathbb Z/2$ in $\pi_{8s+1}^s$, the $(8s+1)$-th stable homotopy group of spheres.
Now regard $\pi_{8s+1}^s = \pi_{8s+1} ((B\...
- 11.9k
12
votes
1
answer
768
views
The multiplication on $THH$ of finite fields
Let $k$ be a finite field, $THH(k)$ its topological Hochschild homology spectrum. For essentially formal reasons, we know that it's an $E_\infty$-algebra over the Eilenberg-Mac Lane spectrum $Hk$, and ...
- 303
20
votes
1
answer
738
views
Can topological cyclic homology compute Picard groups?
Let $K$ be a number field, and $\mathcal{O}_K$ its ring of integers. Then there is an isomorphism
$$K_0(\mathcal{O}_K) \cong \mathbb{Z} \oplus Pic(\mathcal{O}_K)$$
where $Pic(\mathcal{O}_K)$ is the ...
- 4,133
14
votes
2
answers
2k
views
Symplectic K-theory
For a ring $R$ consider symplectic K-theory defined as follows: let $\operatorname{Sp}(R) = \lim_n \operatorname{Sp}_{2n}(R)$, let $\operatorname{ESp}(R)$ be the subgroup generated by elementary ...
- 2,050
14
votes
1
answer
800
views
Is there a category whose isomorphisms are precisely the simple homotopy equivalences?
If we start with the category of finite complexes and continuous maps, and then identify two morphisms iff they are homotopic, we get the homotopy category of finite complexes, and it is trivial to ...
- 18.7k
9
votes
1
answer
711
views
K-theory of the h-cobordism category
I was reading through Kervaire and Milnor's "Groups of Homotopy Spheres", in which the authors begin to compute the groups $\Theta_n$ of h-cobordism classes of homotopy $n$-spheres (with group ...
- 889
13
votes
2
answers
1k
views
Homotopy groups of Fredholm operators
If $X$ is separable complex Hilbert space and $\mathcal{F}$ the topological space of Fredholm operators on $X$, then it is well-known, that
$$ \pi_0(\mathcal{F}) = \mathbb{Z}\, , $$
i.e. the connected ...
- 173
7
votes
2
answers
862
views
Detection of stable homotopy by K-theory spectra
This is primarily a reference request. Does anyone know of any writing about algebraic K-theory spectra picking up elements in the stable homotopy groups of spheres in their Hurewicz image coming from ...
- 10.4k
23
votes
1
answer
949
views
Fundamental theorem of K-theory for loop groups over $\mathbb{F}_1$?
As the title says, I would like to know what the fundamental theorem of algebraic K-theory would say over the field with one element. Recall that the fundamental theorem of K-theory provides a ...
- 17.4k
12
votes
2
answers
794
views
What is the coefficient ring of algebraic K theory of the discrete $\mathbb{C}$?
Ordinary (connective) complex $K$-theory is the algebraic $K$ theory of the topological ring $\mathbb{C}$ with analytic topology. One can also study the $K$ theory of $\mathbb{C}$ with discrete ...
- 7,952
19
votes
2
answers
703
views
Besides $F_q$, for which rings $R$ is $K_i(R)$ completely known?
With the exception of finite fields and "trivial examples", which rings $R$ are such that Quillen's algebraic $K$ groups $K_i(R)$ are completely known for all $i\geq 0$?
Here, by "trivial examples" ...
user1437
29
votes
4
answers
5k
views
Quillen's motivation of higher algebraic K-theory
Almost the same question was already asked on MO Motivation for algebraic K-theory?
However, to my taste, the answers there consider the subject from a more modern point of view.
When I open a book ...
- 21.8k
1
vote
1
answer
331
views
Additivity theorem for algebraic L-theory?
There is an additivity theorem for algebraic K-theory. My question is is there an additivity theorem for algebraic L-theory?
- 411
8
votes
1
answer
486
views
Algebraic K-theory of odd-dimensional spheres
Let $A(X)$ denote the Waldhausen's algebraic K-theory of a space $X$, and let $n$ be odd.
Are the rational homotopy groups of $A(S^n)$ known?
Is the group $\pi_{2k}(A(S^n))$ finite for all ...
- 29.1k
7
votes
0
answers
191
views
Torsion in Whitehead group
Let $\pi$ be a finite group of odd order. What do we know about the torsion subgroup of $Wh(\pi)$? I am especially interested in the $2$-primary part. Is it always trivial?
- 468
19
votes
4
answers
3k
views
Algebraic K-theory and Homotopy Sheaves
Recently, when I was reading the definition of higher algebraic K-theory, I tried to give myself some motivation by looking at derived algebraic geometry. The constructions for algebraic K-theory ...
- 365
11
votes
3
answers
966
views
Waldhausen $K$-theory for $G$-spaces
I would guess that the following is true, and that somebody has worked it out, but I don't recall ever seeing it. Can anyone point me to any literature on it?
Let $G$ be a finite group. We know that ...
- 55.9k
57
votes
2
answers
7k
views
What arithmetic information is contained in the algebraic K-theory of the integers
I'm always looking for applications of homotopy theory to other fields, mostly as a way to make my talks more interesting or to motivate the field to non-specialists. It seems like most talks about ...
- 30.3k
1
vote
1
answer
159
views
homology of $B S^{-1} S$ computation in the proof that $+ = Q$
Let $S$ denote the category of projective (left) $R$-modules with isomorphisms for arrows. We have that
$BS^{-1}S \sim B \text{GL}(R)^+ \times K_0(R)$
In proving this, in Srinivas' algebraic K-...
- 1,105
4
votes
1
answer
885
views
Intuition as to why the K-theory of a ring should be the homotopy theory of an H-space
For an ideal $I \subset R$ with relative K-groups $K_i(R,I)$ we have an exact sequence
$K_2(R) \to K_2(R,I) \to K_2(R/I) \to K_1(R) \to K_1(R,I) \to K_1(R/I)$
$\to K_0(R) \to K_0(R,I) \to K_0(R/I)$
...
- 1,105
1
vote
1
answer
309
views
Simplicial sets from bisimplicial sets, and their realisations.
From a bisimplicial space $T$, one can consider the simplicial spaces $\underline p \mapsto T_{pp} $, $\underline p \mapsto |\underline q \mapsto T_{pq} |$, and $\underline q \mapsto |\underline p \...
- 1,105
3
votes
1
answer
323
views
Path components of a monoidal category form a monoid?
In Grayson's 'Higher Algebraic K-theory II', leading up to the categorical generalisation of the plus construction, he considers $\pi_0(S) = \pi_0(BS)$, where $S$ is a (small, symmetric) monoidal ...
- 1,105
24
votes
3
answers
4k
views
Plus construction considerations.
In order to realise the K-groups of a ring as the homotopy groups of some space associated to that ring, Quillen proposed the following (roughly-sketched) construction:
Recall that $K_1(R) = GL(R)/E(...
- 1,105
7
votes
1
answer
496
views
Waldhausen Additivity in a More General Context
The following arose when I was thinking about a talk at the Midwest Topology Seminar:
Background
I want to consider a generalization of a Waldhausen-like structure on a category $C$ with 0-object $\...
- 18.8k
34
votes
1
answer
2k
views
Is every ''group-completion'' map an acyclic map?
I start with a longer discussion which will result in a precise version of the question. I am puzzled about an issue with the
Quillen plus construction. I have seen outstanding experts being confused ...
- 20.9k
9
votes
1
answer
837
views
K-Theory space of finite abelian groups
Consider the abelian category $\mathsf{finAb}$ of finite abelian groups. It is easy to prove that there is an isomorphism $\mathrm{ord} : K_0(\mathsf{finAb}) \to \mathbb{Q}^+$. Can you give a ...
- 63.1k
2
votes
1
answer
220
views
Cube of cofibrations II
Let $\mathcal{C}$ be a category with cofibrations in the sense of (Waldhausen, Algebraic K-Theory of Spaces) and denote by $F_n(\mathcal{C})$ the category with cofibrations consisting of sequences of $...
- 63.1k
16
votes
2
answers
2k
views
Can anyone explain to me what is an assembly map?
Or can you give me a good place to read about things related to assembly map, besides wikipedia? I am specially interested in the case of algebraic K-theory. Would appreciated if you could provide ...
- 1,598
8
votes
1
answer
499
views
The K-theoretic Farrell-Jones conjecture for cat(0) groups
Is the fibered K-theoretic farrell-jones conjecture true for cat(0)-groups?
- 235
5
votes
1
answer
393
views
A Reference on Multicategories with "Internal Hom"
The multicategory of Waldhausen categories is "enriched over itself": the Hom-set of $k$-exact functors can be given a Waldhausen category structure by letting the morphisms be natural transformations,...
- 1,025
0
votes
0
answers
307
views
A modified version of K-theory for manifolds ?
If $X$ is a compact smooth manifold, $K^{0}(X)$ can be defined as the algebraic $K_{0}$-group of $C^{\infty}(X)$. In order to do that we use the following equivalence relation: we say that two ...
- 505
8
votes
2
answers
395
views
Algorithm to calculate $Wh(G)$ for finitely presented group $G$?
Let $G$ be a finitely presented group.
Are there any algorithm to calculate whitehead group $G$, $Wh(G)$ in terms of presentation of $G$?
- 698
16
votes
2
answers
2k
views
Why was it reasonable to ask what the higher K-groups are?
To say I am a novice in $K$-theory is to overstate my experience with the field. I've been reading the various wiki articles so as to have some preparation before jumping in, and I couldn't answer the ...
- 6,348
55
votes
6
answers
7k
views
Which of Quillen's Papers Should I read?
I just heard that Daniel Quillen passed on. I am not familiar with his work
and want to celebrate his life by reading some of his papers. Which one(s?)
should I read?
I am an algebraic geometer who ...
7
votes
2
answers
580
views
Why does the map $BG\to A(*)$ fail to split?
There is a map $BG \to A(\ast)$ where $BG$ classifies stable spherical fibrations and $A(\ast)$ is
Waldhausen's algebraic $K$-theory of a point. The map is induced by applying Quillen's plus ...
- 18.8k
6
votes
0
answers
312
views
homotopy domination that splits a non-split epimorphism and still wants to be a homotopy equivalence
Can a homotopy domination by a space supporting a free action of $G$ be promoted to a homotopy equivalence with such a space? As stated, this is not a serious question (multiply by an $EG$). But with ...
- 5,575
4
votes
1
answer
328
views
Algebraic K-groups and braids
This is (I think) a reference request:
Are there calculations of any algebraic K-groups for the (group ring of) the Artin braid groups?
- 1,180
37
votes
1
answer
3k
views
Morava on Shafarevich conjecture
$\DeclareMathOperator\Q{\mathbf{Q}}$Jack Morava has some interesting ideas stemming from stable homotopy theory and geometric topology on the Shafarevich Conjecture.
The Shafarevich Conjecture states: ...
- 2,734
-1
votes
2
answers
814
views
Definition for fundamental group (higher homotopy groups) for a category?
How to define homotopy groups in categories as in Quillen's definition for Higher algebraic K-theory: K_i(M)=\pi_{i+1}(BQM, 0), where M is a small category and BQM is the classifying space of QM. ...
- 39
7
votes
2
answers
541
views
(Co-) Homology associated to Waldhausen K-Theory
Waldhausen K-Theory takes as input a Waldhausen category C and produces a spectrum K(C). I would like to know what is known about generalized (co-) homology theories that can be realized by this ...
- 1,273
104
votes
10
answers
24k
views
Motivation for algebraic K-theory?
I'm looking for a big-picture treatment of algebraic K-theory and why it's important. I've seen various abstract definitions (Quillen's plus and Q constructions, some spectral constructions like ...